# Lecture note 37/2008

## Distribution-Valued Analytic Functions - Theory and Applications

### Norbert Ortner and Peter Wagner

**Contact the author:** Please use for correspondence this email.**Submission date: **23. Jan. 2008**Pages: 133****published as: ****Ortner**, **N.** and P. Wagner: Distribution-valued analytic functions : theory and applications

Hamburg : tredition, 2013. - 144 p.

ISBN 978-3-8491-1968-3 **Bibtex****Abstract:**

The aim of this book consists in giving a systematic and general approach
to treating meromorphic distribution-valued functions of
the form
where the "characteristic"
(with also depends
meromorphically on

Let us describe now the contents of the booklet more in detail. Chapter I consists of supplements to the theories of locally convex topological vector spaces and, in particular, of distribution spaces. Hereby, results from the books Schwartz [5], Robertson and Robertson [1], Horváth [4] and Treves [1] are taken for granted and are quoted only. We supplement these basic references by synopses on distributions on hypersurfaces (1.1), on convolution of measures and distributions (1.2, 1.3), on bilinear mappings defined on barrelled spaces (1.4), and on holomorphic functions with values in topological vector spaces (1.5, 1.6).

In Chapter II, the quasihomogeneous distribution-valued functions are defined and their properties (analytic continuation, poles, residues, finite parts) are derived (2.1, 2.2). The structure of quasihomogeneous distributions and of its Fourier transforms is elucidated in 2.5, 2.6. In the remaining sections of Chapter II, the theory is illustrated by several concrete examples originating from the quasihomogeneous polynomials Fundamental solutions of the iterated Cauchy-Riemann operator of the iterated wave operator and, more generally, of the iterated ultrahyperbolic operator are deduced therefrom (Ex. 2.7.3, Prop. 2.4.2, Prop. 2.7.6).

In Chapter III, the convolution with the quasihomogeneous distributions
arising in Chapter II is treated. For this purpose, we define
*weighted* spaces, which generalize the spaces
introduced by L. Schwartz (3.1).
The homogeneous distributions operate on
weighted spaces by convolution, and we obtain
continuity properties in dependence on the regularity of the characteristic
*F* (see 3.2 for
3.3, 3.4 for
3.6 for As application, we describe the
convolution groups of elliptic (3.3), hyperbolic (3.5),
ultrahyperbolic (3.6) and quasihyperbolic operators (3.7).
Finally, the convolution groups of some singular integral operators
are treated in 3.8.